Simplification of discriminant from scale-rotated-ellipse

Percentage Accurate: 24.8% → 83.2%

Time: 2.4min

Alternatives: 3

Speedup: 1693.0×

• 33.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot t\_2}{x-scale}}{y-scale}\\ t\_3 \cdot t\_3 - \left(4 \cdot \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{y-scale}}{y-scale} \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1 (sin t_0))
        (t_2 (cos t_0))
        (t_3
         (/
          (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
          y-scale)))
   (-
    (* t_3 t_3)
    (*
     (*
      4.0
      (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
     (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = sin(t_0);
	double t_2 = cos(t_0);
	double t_3 = ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * Math.PI;
	double t_1 = Math.sin(t_0);
	double t_2 = Math.cos(t_0);
	double t_3 = ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (angle / 180.0) * math.pi
	t_1 = math.sin(t_0)
	t_2 = math.cos(t_0)
	t_3 = ((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale
	return (t_3 * t_3) - ((4.0 * (((math.pow((a * t_1), 2.0) + math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((math.pow((a * t_2), 2.0) + math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale))

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	t_3 = Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale)
	return Float64(Float64(t_3 * t_3) - Float64(Float64(4.0 * Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale)))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (angle / 180.0) * pi;
	t_1 = sin(t_0);
	t_2 = cos(t_0);
	t_3 = ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	tmp = (t_3 * t_3) - ((4.0 * (((((a * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * (((((a * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale));
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(N[(4.0 * N[(N[(N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Initial Program: 24.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{angle}{180} \cdot \pi\\ t_1 := \sin t\_0\\ t_2 := \cos t\_0\\ t_3 := \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot t\_1\right) \cdot t\_2}{x-scale}}{y-scale}\\ t\_3 \cdot t\_3 - \left(4 \cdot \frac{\frac{{\left(a \cdot t\_1\right)}^{2} + {\left(b \cdot t\_2\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot t\_2\right)}^{2} + {\left(b \cdot t\_1\right)}^{2}}{y-scale}}{y-scale} \end{array} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (let* ((t_0 (* (/ angle 180.0) PI))
        (t_1 (sin t_0))
        (t_2 (cos t_0))
        (t_3
         (/
          (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) t_1) t_2) x-scale)
          y-scale)))
   (-
    (* t_3 t_3)
    (*
     (*
      4.0
      (/ (/ (+ (pow (* a t_1) 2.0) (pow (* b t_2) 2.0)) x-scale) x-scale))
     (/ (/ (+ (pow (* a t_2) 2.0) (pow (* b t_1) 2.0)) y-scale) y-scale)))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * ((double) M_PI);
	double t_1 = sin(t_0);
	double t_2 = cos(t_0);
	double t_3 = ((((2.0 * (pow(b, 2.0) - pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((pow((a * t_1), 2.0) + pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((pow((a * t_2), 2.0) + pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	double t_0 = (angle / 180.0) * Math.PI;
	double t_1 = Math.sin(t_0);
	double t_2 = Math.cos(t_0);
	double t_3 = ((((2.0 * (Math.pow(b, 2.0) - Math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	return (t_3 * t_3) - ((4.0 * (((Math.pow((a * t_1), 2.0) + Math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((Math.pow((a * t_2), 2.0) + Math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale));
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	t_0 = (angle / 180.0) * math.pi
	t_1 = math.sin(t_0)
	t_2 = math.cos(t_0)
	t_3 = ((((2.0 * (math.pow(b, 2.0) - math.pow(a, 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale
	return (t_3 * t_3) - ((4.0 * (((math.pow((a * t_1), 2.0) + math.pow((b * t_2), 2.0)) / x_45_scale) / x_45_scale)) * (((math.pow((a * t_2), 2.0) + math.pow((b * t_1), 2.0)) / y_45_scale) / y_45_scale))

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = Float64(Float64(angle / 180.0) * pi)
	t_1 = sin(t_0)
	t_2 = cos(t_0)
	t_3 = Float64(Float64(Float64(Float64(Float64(2.0 * Float64((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale)
	return Float64(Float64(t_3 * t_3) - Float64(Float64(4.0 * Float64(Float64(Float64((Float64(a * t_1) ^ 2.0) + (Float64(b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * Float64(Float64(Float64((Float64(a * t_2) ^ 2.0) + (Float64(b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale)))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	t_0 = (angle / 180.0) * pi;
	t_1 = sin(t_0);
	t_2 = cos(t_0);
	t_3 = ((((2.0 * ((b ^ 2.0) - (a ^ 2.0))) * t_1) * t_2) / x_45_scale) / y_45_scale;
	tmp = (t_3 * t_3) - ((4.0 * (((((a * t_1) ^ 2.0) + ((b * t_2) ^ 2.0)) / x_45_scale) / x_45_scale)) * (((((a * t_2) ^ 2.0) + ((b * t_1) ^ 2.0)) / y_45_scale) / y_45_scale));
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := Block[{t$95$0 = N[(N[(angle / 180.0), $MachinePrecision] * Pi), $MachinePrecision]}, Block[{t$95$1 = N[Sin[t$95$0], $MachinePrecision]}, Block[{t$95$2 = N[Cos[t$95$0], $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(N[(N[(2.0 * N[(N[Power[b, 2.0], $MachinePrecision] - N[Power[a, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision] / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]}, N[(N[(t$95$3 * t$95$3), $MachinePrecision] - N[(N[(4.0 * N[(N[(N[(N[Power[N[(a * t$95$1), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$2), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / x$45$scale), $MachinePrecision] / x$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(N[Power[N[(a * t$95$2), $MachinePrecision], 2.0], $MachinePrecision] + N[Power[N[(b * t$95$1), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] / y$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Alternative 1: 83.2% accurate, 14.9× speedup

Math

\[\begin{array}{l} \\ \frac{{\left(a \cdot b\right)}^{2}}{x-scale \cdot y-scale} \cdot \frac{\frac{-4}{x-scale}}{y-scale} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* (/ (pow (* a b) 2.0) (* x-scale y-scale)) (/ (/ -4.0 x-scale) y-scale)))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (pow((a * b), 2.0) / (x_45_scale * y_45_scale)) * ((-4.0 / x_45_scale) / y_45_scale);
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = (((a * b) ** 2.0d0) / (x_45scale * y_45scale)) * (((-4.0d0) / x_45scale) / y_45scale)
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (Math.pow((a * b), 2.0) / (x_45_scale * y_45_scale)) * ((-4.0 / x_45_scale) / y_45_scale);
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return (math.pow((a * b), 2.0) / (x_45_scale * y_45_scale)) * ((-4.0 / x_45_scale) / y_45_scale)

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64((Float64(a * b) ^ 2.0) / Float64(x_45_scale * y_45_scale)) * Float64(Float64(-4.0 / x_45_scale) / y_45_scale))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = (((a * b) ^ 2.0) / (x_45_scale * y_45_scale)) * ((-4.0 / x_45_scale) / y_45_scale);
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(N[Power[N[(a * b), $MachinePrecision], 2.0], $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(N[(-4.0 / x$45$scale), $MachinePrecision] / y$45$scale), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 26.6%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

3. Simplified 24.3%

   \[\leadsto \color{blue}{\frac{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}{y-scale \cdot x-scale} \cdot \frac{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}{y-scale \cdot x-scale} - 4 \cdot \left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} \cdot \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)} \]
4. Add Preprocessing

5. Taylor expanded in angle around 0 51.0%

   \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
6. Step-by-step derivation

   1. associate-*r/ 51.0%

      \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

   2. *-commutative 51.0%

      \[\leadsto \frac{-4 \cdot \color{blue}{\left({b}^{2} \cdot {a}^{2}\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   3. *-commutative 51.0%

      \[\leadsto \frac{-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)}{\color{blue}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]

   4. unpow2 51.0%

      \[\leadsto \frac{-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)}{\color{blue}{\left(y-scale \cdot y-scale\right)} \cdot {x-scale}^{2}} \]

   5. unpow2 51.0%

      \[\leadsto \frac{-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)}{\left(y-scale \cdot y-scale\right) \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}} \]

   6. swap-sqr 65.5%

      \[\leadsto \frac{-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)}{\color{blue}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}} \]

   7. unpow2 65.5%

      \[\leadsto \frac{-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)}{\color{blue}{{\left(y-scale \cdot x-scale\right)}^{2}}} \]

   8. *-commutative 65.5%

      \[\leadsto \frac{-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)}{{\color{blue}{\left(x-scale \cdot y-scale\right)}}^{2}} \]

7. Simplified 65.5%

   \[\leadsto \color{blue}{\frac{-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
8. Step-by-step derivation

   1. clear-num 65.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{{\left(x-scale \cdot y-scale\right)}^{2}}{-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)}}} \]

   2. inv-pow 65.5%

      \[\leadsto \color{blue}{{\left(\frac{{\left(x-scale \cdot y-scale\right)}^{2}}{-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)}\right)}^{-1}} \]

   3. pow-prod-down 78.6%

      \[\leadsto {\left(\frac{{\left(x-scale \cdot y-scale\right)}^{2}}{-4 \cdot \color{blue}{{\left(b \cdot a\right)}^{2}}}\right)}^{-1} \]

9. Applied egg-rr 78.6%

   \[\leadsto \color{blue}{{\left(\frac{{\left(x-scale \cdot y-scale\right)}^{2}}{-4 \cdot {\left(b \cdot a\right)}^{2}}\right)}^{-1}} \]
10. Step-by-step derivation

    1. unpow-1 78.6%

       \[\leadsto \color{blue}{\frac{1}{\frac{{\left(x-scale \cdot y-scale\right)}^{2}}{-4 \cdot {\left(b \cdot a\right)}^{2}}}} \]

    2. *-commutative 78.6%

       \[\leadsto \frac{1}{\frac{{\left(x-scale \cdot y-scale\right)}^{2}}{-4 \cdot {\color{blue}{\left(a \cdot b\right)}}^{2}}} \]

11. Simplified 78.6%

    \[\leadsto \color{blue}{\frac{1}{\frac{{\left(x-scale \cdot y-scale\right)}^{2}}{-4 \cdot {\left(a \cdot b\right)}^{2}}}} \]
12. Step-by-step derivation

    1. unpow2 78.6%

       \[\leadsto \frac{1}{\frac{\color{blue}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}}{-4 \cdot {\left(a \cdot b\right)}^{2}}} \]

    2. *-commutative 78.6%

       \[\leadsto \frac{1}{\frac{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}{\color{blue}{{\left(a \cdot b\right)}^{2} \cdot -4}}} \]

    3. *-commutative 78.6%

       \[\leadsto \frac{1}{\frac{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}{{\color{blue}{\left(b \cdot a\right)}}^{2} \cdot -4}} \]

    4. times-frac 84.6%

       \[\leadsto \frac{1}{\color{blue}{\frac{x-scale \cdot y-scale}{{\left(b \cdot a\right)}^{2}} \cdot \frac{x-scale \cdot y-scale}{-4}}} \]

    5. *-commutative 84.6%

       \[\leadsto \frac{1}{\frac{x-scale \cdot y-scale}{{\color{blue}{\left(a \cdot b\right)}}^{2}} \cdot \frac{x-scale \cdot y-scale}{-4}} \]

13. Applied egg-rr 84.6%

    \[\leadsto \frac{1}{\color{blue}{\frac{x-scale \cdot y-scale}{{\left(a \cdot b\right)}^{2}} \cdot \frac{x-scale \cdot y-scale}{-4}}} \]
14. Step-by-step derivation

    1. inv-pow 84.6%

       \[\leadsto \color{blue}{{\left(\frac{x-scale \cdot y-scale}{{\left(a \cdot b\right)}^{2}} \cdot \frac{x-scale \cdot y-scale}{-4}\right)}^{-1}} \]

    2. unpow-prod-down 84.8%

       \[\leadsto \color{blue}{{\left(\frac{x-scale \cdot y-scale}{{\left(a \cdot b\right)}^{2}}\right)}^{-1} \cdot {\left(\frac{x-scale \cdot y-scale}{-4}\right)}^{-1}} \]

    3. inv-pow 84.8%

       \[\leadsto \color{blue}{\frac{1}{\frac{x-scale \cdot y-scale}{{\left(a \cdot b\right)}^{2}}}} \cdot {\left(\frac{x-scale \cdot y-scale}{-4}\right)}^{-1} \]

    4. clear-num 84.8%

       \[\leadsto \color{blue}{\frac{{\left(a \cdot b\right)}^{2}}{x-scale \cdot y-scale}} \cdot {\left(\frac{x-scale \cdot y-scale}{-4}\right)}^{-1} \]

    5. inv-pow 84.8%

       \[\leadsto \frac{{\left(a \cdot b\right)}^{2}}{x-scale \cdot y-scale} \cdot \color{blue}{\frac{1}{\frac{x-scale \cdot y-scale}{-4}}} \]

    6. clear-num 84.8%

       \[\leadsto \frac{{\left(a \cdot b\right)}^{2}}{x-scale \cdot y-scale} \cdot \color{blue}{\frac{-4}{x-scale \cdot y-scale}} \]

    7. associate-/r* 84.8%

       \[\leadsto \frac{{\left(a \cdot b\right)}^{2}}{x-scale \cdot y-scale} \cdot \color{blue}{\frac{\frac{-4}{x-scale}}{y-scale}} \]

15. Applied egg-rr 84.8%

    \[\leadsto \color{blue}{\frac{{\left(a \cdot b\right)}^{2}}{x-scale \cdot y-scale} \cdot \frac{\frac{-4}{x-scale}}{y-scale}} \]
16. Add Preprocessing

Alternative 2: 83.2% accurate, 99.6× speedup

Math

\[\begin{array}{l} \\ \frac{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}{x-scale \cdot y-scale} \cdot \frac{-4}{x-scale \cdot y-scale} \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale)
 :precision binary64
 (* (/ (* (* a b) (* a b)) (* x-scale y-scale)) (/ -4.0 (* x-scale y-scale))))

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (((a * b) * (a * b)) / (x_45_scale * y_45_scale)) * (-4.0 / (x_45_scale * y_45_scale));
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = (((a * b) * (a * b)) / (x_45scale * y_45scale)) * ((-4.0d0) / (x_45scale * y_45scale))
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return (((a * b) * (a * b)) / (x_45_scale * y_45_scale)) * (-4.0 / (x_45_scale * y_45_scale));
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return (((a * b) * (a * b)) / (x_45_scale * y_45_scale)) * (-4.0 / (x_45_scale * y_45_scale))

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return Float64(Float64(Float64(Float64(a * b) * Float64(a * b)) / Float64(x_45_scale * y_45_scale)) * Float64(-4.0 / Float64(x_45_scale * y_45_scale)))
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = (((a * b) * (a * b)) / (x_45_scale * y_45_scale)) * (-4.0 / (x_45_scale * y_45_scale));
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := N[(N[(N[(N[(a * b), $MachinePrecision] * N[(a * b), $MachinePrecision]), $MachinePrecision] / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision] * N[(-4.0 / N[(x$45$scale * y$45$scale), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 26.6%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

3. Simplified 24.3%

   \[\leadsto \color{blue}{\frac{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}{y-scale \cdot x-scale} \cdot \frac{\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \left(\sin \left(\frac{angle}{180} \cdot \pi\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}{y-scale \cdot x-scale} - 4 \cdot \left(\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{x-scale}^{2}} \cdot \frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{{y-scale}^{2}}\right)} \]
4. Add Preprocessing

5. Taylor expanded in angle around 0 51.0%

   \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{2} \cdot {b}^{2}}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
6. Step-by-step derivation

   1. associate-*r/ 51.0%

      \[\leadsto \color{blue}{\frac{-4 \cdot \left({a}^{2} \cdot {b}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]

   2. *-commutative 51.0%

      \[\leadsto \frac{-4 \cdot \color{blue}{\left({b}^{2} \cdot {a}^{2}\right)}}{{x-scale}^{2} \cdot {y-scale}^{2}} \]

   3. *-commutative 51.0%

      \[\leadsto \frac{-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)}{\color{blue}{{y-scale}^{2} \cdot {x-scale}^{2}}} \]

   4. unpow2 51.0%

      \[\leadsto \frac{-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)}{\color{blue}{\left(y-scale \cdot y-scale\right)} \cdot {x-scale}^{2}} \]

   5. unpow2 51.0%

      \[\leadsto \frac{-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)}{\left(y-scale \cdot y-scale\right) \cdot \color{blue}{\left(x-scale \cdot x-scale\right)}} \]

   6. swap-sqr 65.5%

      \[\leadsto \frac{-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)}{\color{blue}{\left(y-scale \cdot x-scale\right) \cdot \left(y-scale \cdot x-scale\right)}} \]

   7. unpow2 65.5%

      \[\leadsto \frac{-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)}{\color{blue}{{\left(y-scale \cdot x-scale\right)}^{2}}} \]

   8. *-commutative 65.5%

      \[\leadsto \frac{-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)}{{\color{blue}{\left(x-scale \cdot y-scale\right)}}^{2}} \]

7. Simplified 65.5%

   \[\leadsto \color{blue}{\frac{-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]
8. Step-by-step derivation

   1. clear-num 65.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{{\left(x-scale \cdot y-scale\right)}^{2}}{-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)}}} \]

   2. inv-pow 65.5%

      \[\leadsto \color{blue}{{\left(\frac{{\left(x-scale \cdot y-scale\right)}^{2}}{-4 \cdot \left({b}^{2} \cdot {a}^{2}\right)}\right)}^{-1}} \]

   3. pow-prod-down 78.6%

      \[\leadsto {\left(\frac{{\left(x-scale \cdot y-scale\right)}^{2}}{-4 \cdot \color{blue}{{\left(b \cdot a\right)}^{2}}}\right)}^{-1} \]

9. Applied egg-rr 78.6%

   \[\leadsto \color{blue}{{\left(\frac{{\left(x-scale \cdot y-scale\right)}^{2}}{-4 \cdot {\left(b \cdot a\right)}^{2}}\right)}^{-1}} \]
10. Step-by-step derivation

    1. unpow-1 78.6%

       \[\leadsto \color{blue}{\frac{1}{\frac{{\left(x-scale \cdot y-scale\right)}^{2}}{-4 \cdot {\left(b \cdot a\right)}^{2}}}} \]

    2. *-commutative 78.6%

       \[\leadsto \frac{1}{\frac{{\left(x-scale \cdot y-scale\right)}^{2}}{-4 \cdot {\color{blue}{\left(a \cdot b\right)}}^{2}}} \]

11. Simplified 78.6%

    \[\leadsto \color{blue}{\frac{1}{\frac{{\left(x-scale \cdot y-scale\right)}^{2}}{-4 \cdot {\left(a \cdot b\right)}^{2}}}} \]
12. Step-by-step derivation

    1. clear-num 78.6%

       \[\leadsto \color{blue}{\frac{-4 \cdot {\left(a \cdot b\right)}^{2}}{{\left(x-scale \cdot y-scale\right)}^{2}}} \]

    2. *-commutative 78.6%

       \[\leadsto \frac{\color{blue}{{\left(a \cdot b\right)}^{2} \cdot -4}}{{\left(x-scale \cdot y-scale\right)}^{2}} \]

    3. *-commutative 78.6%

       \[\leadsto \frac{{\color{blue}{\left(b \cdot a\right)}}^{2} \cdot -4}{{\left(x-scale \cdot y-scale\right)}^{2}} \]

    4. unpow2 78.6%

       \[\leadsto \frac{{\left(b \cdot a\right)}^{2} \cdot -4}{\color{blue}{\left(x-scale \cdot y-scale\right) \cdot \left(x-scale \cdot y-scale\right)}} \]

    5. times-frac 84.8%

       \[\leadsto \color{blue}{\frac{{\left(b \cdot a\right)}^{2}}{x-scale \cdot y-scale} \cdot \frac{-4}{x-scale \cdot y-scale}} \]

    6. *-commutative 84.8%

       \[\leadsto \frac{{\color{blue}{\left(a \cdot b\right)}}^{2}}{x-scale \cdot y-scale} \cdot \frac{-4}{x-scale \cdot y-scale} \]

13. Applied egg-rr 84.8%

    \[\leadsto \color{blue}{\frac{{\left(a \cdot b\right)}^{2}}{x-scale \cdot y-scale} \cdot \frac{-4}{x-scale \cdot y-scale}} \]
14. Step-by-step derivation

    1. unpow2 84.8%

       \[\leadsto \frac{\color{blue}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}}{x-scale \cdot y-scale} \cdot \frac{-4}{x-scale \cdot y-scale} \]

15. Applied egg-rr 84.8%

    \[\leadsto \frac{\color{blue}{\left(a \cdot b\right) \cdot \left(a \cdot b\right)}}{x-scale \cdot y-scale} \cdot \frac{-4}{x-scale \cdot y-scale} \]
16. Add Preprocessing

Alternative 3: 35.1% accurate, 1693.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (a b angle x-scale y-scale) :precision binary64 0.0)

C

double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 0.0;
}

Fortran

real(8) function code(a, b, angle, x_45scale, y_45scale)
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: angle
    real(8), intent (in) :: x_45scale
    real(8), intent (in) :: y_45scale
    code = 0.0d0
end function

Java

public static double code(double a, double b, double angle, double x_45_scale, double y_45_scale) {
	return 0.0;
}

Python

def code(a, b, angle, x_45_scale, y_45_scale):
	return 0.0

Julia

function code(a, b, angle, x_45_scale, y_45_scale)
	return 0.0
end

MATLAB

function tmp = code(a, b, angle, x_45_scale, y_45_scale)
	tmp = 0.0;
end

Wolfram

code[a_, b_, angle_, x$45$scale_, y$45$scale_] := 0.0

Derivation

1. Initial program 26.6%

   \[\frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} \cdot \frac{\frac{\left(\left(2 \cdot \left({b}^{2} - {a}^{2}\right)\right) \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right) \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)}{x-scale}}{y-scale} - \left(4 \cdot \frac{\frac{{\left(a \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{x-scale}}{x-scale}\right) \cdot \frac{\frac{{\left(a \cdot \cos \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle}{180} \cdot \pi\right)\right)}^{2}}{y-scale}}{y-scale} \]

3. Simplified 25.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\cos \left(\frac{angle \cdot \pi}{180}\right) \cdot \frac{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}{x-scale \cdot y-scale}, \cos \left(\frac{angle \cdot \pi}{180}\right) \cdot \frac{\left({b}^{2} - {a}^{2}\right) \cdot \left(2 \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}{x-scale \cdot y-scale}, \frac{{\left(a \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{{y-scale}^{2}} \cdot \left(\frac{{\left(a \cdot \sin \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2} + {\left(b \cdot \cos \left(\frac{angle \cdot \pi}{180}\right)\right)}^{2}}{{x-scale}^{2}} \cdot -4\right)\right)} \]
4. Add Preprocessing

5. Taylor expanded in b around 0 25.2%

   \[\leadsto \color{blue}{-4 \cdot \frac{{a}^{4} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} + 4 \cdot \frac{{a}^{4} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}}} \]
6. Step-by-step derivation

   1. distribute-rgt-out 25.2%

      \[\leadsto \color{blue}{\frac{{a}^{4} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot \left(-4 + 4\right)} \]

   2. metadata-eval 25.2%

      \[\leadsto \frac{{a}^{4} \cdot \left({\cos \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2} \cdot {\sin \left(0.005555555555555556 \cdot \left(angle \cdot \pi\right)\right)}^{2}\right)}{{x-scale}^{2} \cdot {y-scale}^{2}} \cdot \color{blue}{0} \]

   3. mul0-rgt 32.5%

      \[\leadsto \color{blue}{0} \]

7. Simplified 32.5%

   \[\leadsto \color{blue}{0} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024096 
(FPCore (a b angle x-scale y-scale)
  :name "Simplification of discriminant from scale-rotated-ellipse"
  :precision binary64
  (- (* (/ (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI))) (cos (* (/ angle 180.0) PI))) x-scale) y-scale) (/ (/ (* (* (* 2.0 (- (pow b 2.0) (pow a 2.0))) (sin (* (/ angle 180.0) PI))) (cos (* (/ angle 180.0) PI))) x-scale) y-scale)) (* (* 4.0 (/ (/ (+ (pow (* a (sin (* (/ angle 180.0) PI))) 2.0) (pow (* b (cos (* (/ angle 180.0) PI))) 2.0)) x-scale) x-scale)) (/ (/ (+ (pow (* a (cos (* (/ angle 180.0) PI))) 2.0) (pow (* b (sin (* (/ angle 180.0) PI))) 2.0)) y-scale) y-scale))))